Memory-type variance estimators using exponentially weighted moving average statistic in presence of measurement error for time-scaled surveys

The present study suggested memory-type ratio and product estimators for variance estimation in the presence of measurement errors. We applied the exponentially weighted moving averages statistic which simultaneously utilizes the current and prior information for better estimation in surveys based on the time-scale. The expressions of approximate mean square errors of memory-type estimators are derived using Taylor series up to first order. Mathematical conditions are also obtained for which the suggested memory-type ratio and product estimators perform better than the conventional ratio and product estimators. The efficiency of the proposed estimators is observed using an extensive simulation study in the presence of measurement errors. A real data application is also carried out to support the mathematical expressions. From the results, it is shown that the use of prior sample information significantly increased the efficiency of the proposed estimators.


Introduction
Surveys are relatively inexpensive and useful to describe the characteristics of large populations.It plays a vital role in many disciplines to gather the information for the estimation of unknown parameters of the population.Many surveys have been designed and administered in many modes and repeated under the similar approach with equal time space all over the globe.For instance, the Australian Bureau of Statistics and the National Bureau of Statistics of China conducted health, economic and agricultural surveys annually, quarterly, and monthly.Every year, the Pakistan Bureau of Statistics conducts the Labor Force Survey (LFS) to measure the different statistics of Pakistan.Bureau of Statistics, Punjab, has been conducting Multiple Indicator Cluster Surveys (MICS) regularly since 2004.
In survey sampling, statisticians usually effort to enhance the performance of the estimators by using the auxiliary information correlated with the study variable.In practice, the researchers disposed to get information on many variables rather than the study variable only.Ratio-type estimators are considered to be good example for the situation having a positive correlation between the auxiliary variable and the study variable whereas product-type estimators are assumed to be better for the negatively correlated variables.
The estimation of variance is important when the variability control is difficult in its application such as in agricultural research, experiments in biosciences, manufacturing industries, and pharmaceutical laboratories.The estimation of population variance in the study variable also received significant interest from various authors such as [1][2][3][4][5][6][7][8][9][10][11].
Measurement errors (MEs) are defined as the difference between observed observations and the true observations of the study variable [12].It has severe effects on the estimation of population parameters in terms of increased bias and variation.It is essential to study the impact of ME to develop better estimation techniques and to get more reliable and efficient estimates of the underlying population parameters.Many authors have studied the effect of ME along with the estimation of population parameters.One may refer to [13][14][15][16][17][18].Several researchers like [19][20][21][22][23][24] contribute to the variance estimation of the concerned variable in the presence of ME.
Initially, [25] introduced the idea of EWMA statistic and recycled the current and previous information to observe the change in the process mean.The memory-type ratio and product estimators for mean estimation were suggested by [26] using EWMA statistic under simple random sampling without replacement (SRSWOR).Similarly, [27] proposed the memory type ratio and product estimators for population mean using stratified sampling.
The EWMA statistic for population variance estimation of the study variable with 0�λ�1 is defined as: where ŝ2 yt is the current sample variance and λ is the weighted parameter or smoothing constant of the given observations.As λ increases from 0 to 1, the larger weight is given to current information and simultaneously smaller weight to the past observations.For λ = 1, all the weight is received by current information and EWMA statistic performs the same as the conventional variance estimator.Here, t represents the sample numbers and V t−1 reveals the past observations.The initial value V 0 is taken as one or the average value of the prior sample.
In this study, memory-type ratio and product estimators are suggested in the presence of ME for the time-scaled surveys.After a brief discussion on surveys, auxiliary information, variance estimation, ME, and EWMA statistics, the rest of the paper is as follows.Section 2 is based on the sampling scheme, notations, and conventional estimators under ME.Proposed estimators along with the derivation of approximate mean square error (MSE) are given in Section 3. Mathematical comparisons between the proposed memory-type estimators and the corresponding conventional estimators are given in Section 4. A simulation study is conducted in Section 5 to assess the performance of the proposed estimators over the conventional estimators.Real data application is also used in Section 6 to validate the findings of the mentioned estimators obtained through simulations.Final remarks are given in Section 7.

Sampling procedure and conventional estimators in the presence of ME
Let Y and X be the study variable and the auxiliary variable respectively defined on N identifiable but distinct units of a population U = {U 1 ,U 2 ,U 3 . ..,UN }.Let n pairs of observations be obtained by using SRSWOR on two variables Y and X.Consider a situation where both Y and X variables are observed with some considerable error.Let for the i th sampling unit (i=1, 2 . ..n), y i and x i be the observed instead of true observations Y i and X i .The MEs may be defined as where u i and v i are the associated MEs with constant or zero mean and known variances S 2 U and S 2 V respectively.Following [28], it is assumed that the errors u i and v i are independent of each other and also independent of Y i and X i .This implies that COV(X, Y) It is further assumed that the finite population correction is neglected.The expected values of s 2 y and s 2 x in the presence of MEs are defined as and It is assumed that the error variances, S 2 U ad S 2 V of the study and the auxiliary variables are known.The unbiased estimator of S 2 y and S 2 x are respectively given by and Let the sampling errors for the study and auxiliary variables are defined as where z = Y, X, U, V.
The derivations of the above error terms are given in the S1 Appendix.
The usual sample variance estimator with the equation of its variance in the presence of ME is and The classical ratio estimator given by [29] under ME is defined as The expression MSE for ratio estimator under ME is Similarly, the product estimator along with the expression of MSE in the presence of ME is given as and

Proposed memory-type estimators
In this section, we processed the EWMA statistic for both study and the auxiliary variables to propose the memory-type estimators in the presence of ME.The EMWA statistic for the study variable is defined as Similarly, the EWMA statistic for the auxiliary variable x is defined as A pilot study must have reliable results before conducting every survey.The initial values of V t and Ŵ t are taken as average values estimated from the pilot study or taken as one.The memory-type ratio and product estimators based on EWMA statistic using ME are defined as where S 2 x is assumed to be the known population variance of the auxiliary variable.To derive the approximate mean square errors of both memory-type estimators, let us define the following error terms as We may rewrite the memory-type ratio estimator in terms of sampling errors as Simplifying and applying Taylor series up to the second-order on Eq (2), we have Shortening and applying expectation on both sides of Eq (3), we have the MSE of tM rt as The MSEð tM rt Þ in the presence of ME as Similarly, the MSE of the memory-type product estimator is The MSEð tM pt Þ in the presence of ME as

Mathematical comparisons
The efficiency comparisons of the proposed memory-type estimators over the conventional estimators in terms of MSE are i.The proposed memory-type ratio estimator will perform better than the conventional ratio estimator if ii.The proposed memory-type product estimator will perform better than the conventional product estimator if As λ value ranges from 0 to 1. Therefore, under the condition, λ<1, the proposed memory-type estimator would always be more efficient than the conventional ratio estimator in presence of ME.But if we take λ = 1, the performance of both memory-type and usual estimators will be equal.

Simulation study
An extensive simulation study is conducted to evaluate the performance of the proposed memory-type estimators in the presence of ME.The MSE and relative efficiency (RE) of the proposed and existing estimators are computed by using the following formulas with 50,000 replications as and where t i ¼ s 2 y ; t r ; t M rt ; and t M pt : The results of memory-type ratio and product estimators are computed at different levels of correlation ρ YX (= 0.80,0.85,0.90 and 0.95) and weight parameter λ(= 0.10,0.20,0.50,0.75)by using the following algorithm: 1.A population of size N = 5000 is generated using multivariate normal distribution having 2. Different values have opted for weight constant λ.
4. Fifty thousand iterations are used to compute the proposed estimator under ME from the samples obtained in Step 3.

5.
The MSE and RE of each estimator are calculated for each sample size and presented in Tables 1-4.
From the numerical findings summarized in Tables 1 and 2, it is observed that the MSEs of the memory-type ratio estimator are less than the MSEs of the conventional ratio estimator in the presence of ME.Equivalently, the REs presented in Tables 3 and 4 shown that the proposed ratio estimator is more efficient than the conventional ratio estimator.The effect of ρ YX is observed as it increases up to 1, the MSE decreases and the efficiency of the proposed memory-type ratio estimator increases which revealed that the use of the auxiliary variable, correlated with the survey variable may enhance the efficiency of the proposed estimator.The role of sample size is significant in the efficiency of the proposed estimator.The MSEs of the proposed estimator decrease as the sample size increases.Similarly, the weight λ assigned to the current and previous values has the great impact on the efficiency of the proposed memory- type ratio estimator as shown in Tables 3 and 4. It is further noticed that, when the value of λ goes down, the larger weight is assigned to previous values and hence increases the efficiency of the proposed estimators.For λ = 1, the efficiency of proposed estimators based on EWMA statistic will be as good as conventional estimators.
To check the efficiency of the memory-type product estimator, we generated a bivariate population of size 1000 having negative correlation between the concerned variable and the auxiliary variable.The detail of the population parameter is The results of MSE and RE of proposed and conventional ratio and product estimators are summarized in Tables 5 and 6 displayed that the product type estimators perform better than the ratio estimators in the presence of ME for the population having a negatively correlated data with the study variable.Moreover, the proposed memory-type product estimator found to be more efficient than the conventional product estimator on various sample sizes and λ.The MSEs of the product eestimator decreases as the sample size increases from 50 to 500.

Real-data application
A real data is taken from [30] to confirm the application of memory-type ratio estimator proposed in this paper.The data set contains the true consumption expenditure as study variable Y and true income as an auxiliary variable X whereas the variables contaminated with ME are taken y as measured consumption expenditure and x as measured income with the respective population parameters are N = 10, μ y = 127, μ x = 170, S 2 y ¼ 1420; S 2 x ¼ 3666:67; S 2 U ¼ 36; S 2 V ¼ 41:24; and ρ yx = 0.964.By considering regular time intervals, twenty samples, each of size 4, with λ = 0.2 are chosen by using SRSWOR.Here, we only considered the conventional and proposed memory-type ratio estimators as the correlation coefficient between the study and the auxiliary variables is positive.The results of all the estimators (except the product-type estimators) in the presence and absence of ME are computed and reported in Table 7.It is clear from the numerical findings given in Table 8, that the proposed memory-type ratio estimator under ME is more efficient than the conventional ratio estimator in terms of MSE and RE.

Conclusion
In this paper, we have suggested memory-type ratio and product estimators based on EWMA statistics for estimating the population variance in the presence and absence of ME.The proposed estimators use the current as well as previous sample information.The reported results show that the proposed estimators estimated the population variance more efficiently than the conventional estimators.The ratio (memory-type and conventional) estimators perform well for the populations having a positive correlation between the study and the auxiliary variables, whereas the product (memory-type and conventional) estimators perform better for negatively correlated populations.Further, the REs of the proposed estimators increase, and the MSEs decrease as the sample size increases.The results of real data application confirm the findings obtained from the simulation.Consequently, based on numerical results, it is suggested that the proposed memory-type ratio and product estimators based on EWMA statistics are superior and more useful for estimating population variance for the time-scaled surveys.
Current manuscript uses only one auxiliary variable to estimate the population variance.This approach can be extended to estimate various population characteristics using two or more auxiliary variables in the presence of ME and non-response.